Section 7.1: Antiderivatives

Up until this point in the course, we have mainly been concerned with one general type of problem: Given a function f(x), ﬁnd the derivative f 0(x). In this section, we will be concerned with solving this problem in the reverse direction. That is, given a function f(x), ﬁnd a function F (x) such that F 0(x) = f(x).

This is called antidiﬀerentiation, and any such function F (x) will be called an antiderivative of f(x).

DEFINITION: If F 0(x) = f(x), then F (x) is called an antiderivative of f(x).

Example: If f(x) = 20x4, ﬁnd a function F (x) such that F 0(x) = f(x).

Is the above answer unique?

Write down a formula for every antiderivative of f(x) = 20x4.

Examples:

1. Find an antiderivative for the following functions.

(a) f(x) = x4 2

(b) f(x) = x15

1 (c) f(x) = x2

√ (d) f(x) = x

Indeﬁnite Integral:

INDEFINITE INTEGRAL: If F 0(x) = f(x), we can write this using indeﬁnite integral notation as Z f(x) dx = F (x) + C, where C is a constant called the constant of integration.

Note: We call the family of antiderivatives F (x) + C the general antiderivative of f(x). 3

Antiderivatives and Indeﬁnite Integrals:

Let us develop some rules for evaluating indeﬁnite integrals (antiderivatives). The ﬁrst will be the power rule for antiderivatives. This is simply a way of undoing the power rule for derivatives.

POWER RULE: For any real number n 6= 1, Z xn+1 xn dx = + C, n + 1 or Z 1 xn dx = · xn+1 + C. n + 1

1. Find the following indeﬁnite integrals.

Z (a) x5 dx

Z 1 (b) √ dx 3 x 4

CONSTANT MULTIPLE AND SUM OR DIFFERENCE RULES: If all indicated integrals exist, Z Z k · f(x) dx = k f(x) dx,

for any real number k, and Z Z Z (f(x) ± g(x)) dx = f(x) dx ± g(x) dx.

The constant multiple and sum or difference rules, combined with the power rule for antiderivatives, will allow us to find the antiderivatives of a plethora of different combinations of power functions.

Examples:

2. Find the following indeﬁnite integrals.

Z 4 (a) 17x5/2 − + 5 dx x3

Z (b) 4x(x2 + 2x) dx 5

Indeﬁnite Integrals of Exponential Functions:

Exercise: Given your knowledge of derivatives, try to ﬁnd the following indeﬁnite integrals without a formula.

Z (a) ex dx

Z (b) ekx dx

Z (c) 2x dx

Z (d) 3kx dx

This kind of thinking leads us to easily determine the proper formulas for the indeﬁnite integrals of exponential functions. 6

INDEFINITE INTEGRALS OF EXPONENTIAL FUNCTIONS:

Z • ex dx = ex + C

Z ekx • ekx dx = + C k

Z ax • ax dx = + C ln(a)

Z akx • akx dx = + C k ln(a)

In the above formulas, we are assuming that k 6= 0 and that a > 0 with a 6= 1.

Examples:

3. Find the following indeﬁnite integrals.

Z (a) 3ex dx

Z (b) 4e−0.2x dx 7 Z (c) (e2u + 4u) du

There is one ﬁnal antiderivative property we need. What happens if we attempt to use the power Z rule to ﬁne x−1 dx?

1 INDEFINITE INTEGAL OF : x

Z Z 1 x−1 dx = dx = ln |x| + C. x 8

Examples:

4. Find the following indeﬁnite integrals.

Z 3 (a) dx x

Z 9 (b) − 3e−0.4x dx x

√ Z x + 1 (c) √ dx 3 x 9

Speciﬁc Antiderivatives (Solving for C):

Examples:

5. If f(x) = e2x − 9x2 and F (0) = 4, ﬁnd F (x), the speciﬁc antiderivative of f(x).

6. Find the cost function if the marginal cost function is 1 C0(x) = x + x2 and 2 units cost $5.50. 10

7. Find the demand function if the marginal revenue is √ R0(x) = 500 − 0.15 x.